The use of adapted Besov spaces builds on recent work concerning invariant spaces for Schrödinger and wave equations. Indeed, modulation spaces, invariant spaces for Schrödinger propagators, have been used extensively as spaces of initial data for nonlinear Schrödinger equations (see [1]}, [2]}, [3]}, [4]} and references therein). On the other hand, a scale \((\mathcal {H}^{p}_{FIO}(\mathbb {R}^{d}))_{1\le p\le \infty }\)  of Hardy spaces for Fourier integral operators (FIOs) was introduced in [5]} by Hassell, Portal and the first author. This work in turn generalizes the case \(p=1\)  due to Smith [6]}, which predates [5]} by decades. The Hardy spaces for FIOs are invariant under half-wave propagators and more general FIOs, and they satisfy the Sobolev embeddings
\(W^{s(p),p}(\mathbb {R}^{d})\subseteq \mathcal {H}^{p}_{FIO}(\mathbb {R}^{d})\subseteq W^{-s(p),p}(\mathbb {R}^{d})\) 
